Problem 43 - Algebra and Trigonometry

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=\cos \left(x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The amplitude is 1, the period is \(2\pi\), and the phase shift is \(-\pi/2\).

Step 1: Find the Amplitude

The amplitude of a cosine function is the absolute value of the coefficient of the cosine function. In this case, it is \(|\text{coefficient of } \cos(x-\frac{\pi}{2})|\), i.e., 1.

Step 2: Calculate the Period

The period of the function is determined by dividing the regular period of the function by the absolute value of the coefficient of x, i.e., \(\text{period of } \cos(x-\frac{\pi}{2})\) = \(2\pi / |\text{coefficient of } x|\) = \(2\pi / 1 = 2\pi\).

Step 3: Determine the Phase Shift

The phase shift is the value by which each point of the basic cosine function has been translated horizontally. In this case, it is \(-\pi/2\). Thus, each point of the basic cosine function \(y=\cos(x)\) has been shifted to the left by \(\pi/2\).

Step 4: Sketch the Function

You start by drawing the basic cosine function, then shift each point to the left by \(\pi/2\) units. Note that the amplitude (1) means the function will oscillate between -1 and 1 and it will complete a full cycle every \(2\pi\) units.